📚 I – Planes | 平面
In the IB Mathematics Higher Level course, planes in three-dimensional space form a fundamental part of the vectors and geometry topic. Understanding how to represent planes using vector and Cartesian equations, and performing calculations such as distances, angles, and intersections, is essential for success in the exams. This article systematically reviews all key concepts and problem-solving techniques related to planes.
在 IB 数学高级课程中，三维空间中的平面是向量与几何部分的基石。掌握如何用向量方程和笛卡尔方程表示平面，以及进行距离、夹角和相交等计算，对考试成功至关重要。本文将系统梳理与平面相关的所有核心概念与解题技巧。
1. Introduction to Planes in 3D Space | 三维空间中的平面简介
A plane in three-dimensional space is a flat, two-dimensional surface that extends infinitely. It can be uniquely determined by a point and two non-parallel direction vectors, or by a point and a normal vector perpendicular to the plane. In IB Mathematics, you are expected to work with planes in both vector and Cartesian forms.
三维空间中的平面是一个平坦、无限延伸的二维曲面。它可以由一个点与两个不共线的方向向量唯一确定，或者由一个点和一个垂直于平面的法向量确定。在 IB 数学中，你需要同时处理平面的向量形式和笛卡尔形式。
2. Vector Equation of a Plane | 平面的向量方程
The vector equation of a plane can be written as r = a + λb + μc, where a is the position vector of a point on the plane, and b and c are two non-parallel direction vectors lying in the plane. The parameters λ and μ can take any real values. Alternatively, the scalar product form is r · n = a · n, where n is a normal vector to the plane.
平面的向量方程可写为 r = a + λb + μc，其中 a 是平面上一点的位置向量，b 和 c 是平面上两个不共线的方向向量，参数 λ 和 μ 可取任意实数。另一种标量积形式为 r · n = a · n，其中 n 是平面的法向量。
For example, a plane passing through point A(1,2,3) with direction vectors b = i + j and c = j + k can be expressed as r = (1,2,3) + λ(1,1,0) + μ(0,1,1).
例如，过点 A(1,2,3) 且方向向量为 b = i + j 和 c = j + k 的平面可表示为 r = (1,2,3) + λ(1,1,0) + μ(0,1,1)。
3. Scalar or Cartesian Equation | 标量（笛卡尔）方程
plane is given by ax + by + cz = d, where (a, b, c) are the components of a normal vector n perpendicular to the plane, and d is a constant. Here, d = a · n for any known point a on the plane. This form is extremely useful for quickly identifying the normal vector and computing distances.
平面的标量方程（也称笛卡尔方程）为 ax + by + cz = d，其中 (a, b, c) 是垂直于平面的法向量 n 的分量，d 为常数。对于平面上任意已知点 a，有 d = a · n。该形式非常便于快速识别法向量及计算距离。
To find the Cartesian equation from a vector form r = a + λb + μc, compute the normal vector n = b × c and then use r · n = a · n.
若要从向量形式 r = a + λb + μc 获得笛卡尔方程，先计算法向量 n = b × c，然后代入 r · n = a · n 即可。
4. Converting Between Vector and Cartesian Forms | 向量形式与笛卡尔形式之间的转换
nding one point on the plane and two non-parallel direction vectors. One method is to set two variables to zero to solve for the third, giving a point. Then set one variable as a parameter and express the others accordingly, obtaining direction vectors parallel to the plane. Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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